Asymptotic Behavior at Infinity for Green's Functions of First Order Systems with Characteristics of Nonunlform Multiplicity

Uniform asymptotic profiles of stationary wave propagation in perturbed two‐layered media

“…For the former, we employ the usual stationary phase method as in the proof of (13). For the latter, noting (27), we repeat the same argument as in the proof for (12). Now, we prove (26).…”

“…Now, we turn to the proof of (12). Noting (i) to (iv), using H j (Θ 2 ; x)(j = 1, 2) and integrating by parts, we have…”

“…where Cs are positive constants independent of (x, y, ) satisfying |x| ≧ r 0 , cos 0 < 3 ≦ 1, | | ≦ m and 3∕a ≦ ≦ 2b. Thus we obtain (12). For 0 < 3 < cos 0 , we decompose J (+) 0 (x, , ) into the integrals on the neighborhoods of 3 = 1, 3 = cos ca , 3 = 0, 3 = cos( − ca ) and 3 = −1.…”

Singularities of the Scattering Kernel for Two Convex Obstacles

“…To those points we assign the numbers , these theorems corresponding to Theorem 1 and Theorem 2 have been proved by Nakamura and Soga [10] under the slightly different assumptions. The main tasks in the proof of Theorem 2 are to show that there exist actually the broken rays with the properties (i) 9 (ii) and (iv) stated in Theorem 2. In [10] they have been proved the existence of such rays in the case of two disjoint disks in J2 2 , and that the proof in R z can be reduced to that in R 2 when both 0 1 and 0 2 are balls.…”

Body waves in a weakly anisotropic medium-II.Swaves from a centre of expansion andPwaves from a centre of rotation

“…The solution can be asymptotically evaluated in the far‐field area (e.g. Musgrave 1970; Matsumura 1976; Yeatts 1984; Babich 1998). However, in the general case these results involve some characteristics of a geometrical nature, such as curvature of the slowness surface, for which analysis and computation are far from trivial (see e.g.…”